Varieties and Schemes Revisited

In Basics of Algebraic Geometry we introduced the idea of varieties and schemes as being kinds of “shapes” defined by polynomials (or rings, more generally) in some way. In this post we discuss the definitions of these concepts in more technical detail, and introduce other important concepts related to algebraic geometry as well.

I. Preliminaries: Affine Space, Algebraic Sets and Ringed Spaces

Affine $n$-space, written $\mathbb{A}^{n}$, is the set of all $n$-tuples of elements of a field $k$, i.e.

$\displaystyle \mathbb{A}^{n}=\{(a_{1},...,a_{n})|a_{i}\in k \text{ for }1\leq i\leq n\}$.

An algebraic set is a subset of $\mathbb{A}^{n}$ that is the zero set $Z(T)$ of some set $T$ of polynomials, i.e. $Y=Z(T)$, where

$\displaystyle Z(T)=\{P\in \mathbb{A}^{n}|f(P)=0 \text{ for all } f\in T\}$.

Intuitively, we want to define a “variety” as some kind of space which “locally” looks like an irreducible algebraic set. “Irreducible” means it cannot be expressed as the union of other algebraic sets. However, we want to think of a variety as more than just a space; we want to think of it as a space with things (namely functions) “living on it”. This leads us to the notion of a ringed space.

A ringed space is simply a pair $(X,\mathcal{O}_{X})$, where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf (see  Sheaves) of rings on $X$. A morphism of ringed spaces from $(X,O_{X})$ to $(Y,O_{Y})$ is given by a continuous map $f: X\rightarrow Y$ and a morphism of sheaves of rings $f^{\#}: \mathcal{O}_{Y}\rightarrow f_{*}\mathcal{O}_{X}$.

Recall that a morphism of sheaves of rings $\varphi:\mathcal{F}\rightarrow \mathcal{G}$ for sheaves of rings $\mathcal{F}$ and $\mathcal{G}$ on X is given by a morphism of rings $\varphi(U): \mathcal{F}(U)\rightarrow \mathcal{G}(U)$ for every open set $U$ of $X$ such that for $V\subseteq{U}$ we have $\rho_{U,V}\circ\varphi(U)=\varphi(V)\circ\rho'_{U,V}$, where $\rho_{U,V}$ and $\rho'_{U,V}$ are the restriction maps of $\mathcal{F}$ and $\mathcal{G}$.

We might as well mention locally ringed spaces here, since they will be used to define the concept of schemes later on:

A locally ringed space is a ringed space $(X,\mathcal{O}_{X})$ such that for each point $P$ of $X$, the stalk $\mathcal{O}_{X,P}$ is a local ring (see Localization). A morphism of locally ringed spaces from $(X,O_{X})$ to $(Y,O_{Y})$ is given by a continuous map $f: X\rightarrow Y$ and a morphism of sheaves of rings $f^{\#}: \mathcal{O}_{Y}\rightarrow f_{*}\mathcal{O}_{X}$ such that $(f_{P}^{\#})^{-1}(\mathfrak{m}_{X,P})=\mathfrak{m}_{Y,f(P)}$ for all $P$ where $f_{P}^{\#}: \mathcal{O}_{Y,f(P)}\rightarrow \mathcal{O}_{X,P}$ is the map induced on the stalk at $P$.

II. Varieties in Three Steps:  Affine Varieties, Prevarieties, and Varieties

We now set out to accomplish our goal of defining “varieties” as spaces that locally look like irreducible algebraic sets. We first start with a ringed space that just “looks like” an irreducible algebraic set:

An affine variety is a ringed space $(X,\mathcal{O}_{X})$ such that $X$ is irreducible, $O_{X}$ is a sheaf of $k$-valued functions, and $X$ is isomorphic to an irreducible algebraic set in $\mathbb{A}^{n}$.

Next, we define a more general kind of ringed space, that is required to look like an irreducible algebraic set only “locally”:

A prevariety is a ringed space $(X,\mathcal{O}_{X})$ such that $X$ is irreducible, $O_{X}$ is a sheaf of $k$-valued functions, and there is a finite open cover $U_{i}$ such that $(U_{i},\mathcal{O}_{X}|_{U_{i}})$ is an affine variety for all $i$.

We are almost done. However, there is one more nice property that we would like our varieties to have. A topological space $X$ is said to have the Hausdorff property if two distinct points always have two disjoint neighborhoods. With the Zariski topology this is almost always impossible; however there is an analogous notion which is satisfied if the image of the “diagonal morphism” which sends the point $P$ in $X$ to the point $(P,P)$ in $X\times X$ is closed in $X\times X$. There is an analogous notion of “product” in algebraic geometry; therefore, we can define the concept of variety as follows:

A variety is a prevariety $X$ such that the diagonal morphism is closed in $X\times X$. In the rest of this post, we will refer to this property as the “algebro-geometric” analogue of the Hausdorff property.

III. Schemes

We now define the concept of schemes, which, as we shall show in the next section, generalize the concept of varieties, i.e. varieties are just a special case of schemes. Inspired by the correspondence between the maximal ideals of the “ring of polynomial functions” (with coefficients in an “algebraically closed field” like the complex numbers) of an algebraic set and the points of the algebraic set mentioned in Basics of Algebraic Geometry, we go further and consider a ringed space whose underlying topological space has points corresponding to the prime ideals of a ring (which is not necessarily a ring of polynomials – we might even consider, for example, the ring of ordinary integers $\mathbb{Z}$, or the ring of integers of an algebraic number field –  see Algebraic Numbers).

The spectrum (note that the word “spectrum” has many different meanings in mathematics, and this particular usage is different, say, from that in Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) of a ring is a  locally ringed space $(\text{Spec}(A)),\mathcal{O}$, where $\text{Spec}(A)$ is the set of prime ideals of $A$ equipped with the Zariski topology, and $\mathcal O$ is a sheaf on $\text{Spec}(A)$ given by defining $\mathcal{O}(U)$ to be the set of functions $s:U\rightarrow \coprod_{\mathfrak{p}\in U}A_{\mathfrak{p}}$, such that $s(\mathfrak{p})\in A_\mathfrak{p}$ for each $\mathfrak{p}\in U$, and such that for each $\mathfrak{p}\in U$, there is an open set $V\subseteq U$ containing $\mathfrak{p}$ and elements $a,f\in A$ such that for each $\mathfrak{q}\in V$, $f\notin \mathfrak{q}$, and $s(\mathfrak{q})=a/f$ in $A_{\mathfrak{q}}$.

We now proceed to define schemes, closely mirroring how we defined varieties earlier:

An affine scheme is a locally ringed space $(X,\mathcal{O}_{X})$ that is isomorphic as a locally ringed space to the spectrum of some ring.

A scheme is a locally ringed space $(X,\mathcal{O}_{X})$ where every point is contained in some open set $U$ such that $U$ considered as a topological space, together with the restricted sheaf $\mathcal{O}_{X}|_{U}$, is an affine scheme. A morphism of schemes is a morphism as locally ringed spaces.

Finally, to complete the analogy with varieties, we refer to schemes which have the (analogue of the) Hausdorff property as separated schemes.

Note: In some of the (mostly older) literature, what we refer to as schemes in this post are instead referred to as preschemes, in analogy with prevarieties. What they call a scheme is what we refer to as a separated scheme, i.e. a scheme possessing the Hausdorff property. I have no idea at the moment as to why this rather nice terminology was changed, but in this post we stick with the modern convention.

IV. Prevarieties and Varieties as Special Kinds of Schemes

We now discuss varieties as special cases of schemes. First we need to define what properties we would like our schemes to have, in order to fit with how we described varieties earlier (as ringed spaces which locally look like irreducible spaces defined by polynomials). Therefore, we have to mimic certain properties of polynomial rings.

We first note that polynomials over a field are finitely generated algebras over some field $k$. A scheme is said to be of finite type over the field $k$ if the affine open sets are each isomorphic to the spectrum of some ring which is a finitely generated algebra over $k$. More generally, given a morphism of schemes $X\rightarrow Y$, there is a concept of $X$ being a scheme of finite type over $Y$, but we will leave this to the references for now.

Next we note that polynomials over a field are integral domains. This means that whenever there are two polynomials $f$ and $g$ with the property that $fg=0$, then either $f=0$ or $g=0$. A scheme is integral if each the affine open sets are each isomorphic to the spectrum of some ring which is an integral domain. An equivalent condition is for the scheme to be irreducible and reduced (this means that the ring specified above has no nilpotent elements, i.e. elements where some power is equal to zero).

We therefore redefine a prevariety as an integral scheme of finite type over the field $k$. As with the earlier definition, a variety is a prevariety with the (analogue of the) Hausdorff property (i.e. an integral separated scheme of finite type over $k$).

V. Conclusion

In conclusion, we have started with essentially the same ideas as the “analytic geometry” of Pierre de Fermat and Rene Descartes, familiar to high school students everywhere, used to describe shapes such as lines, circles, conics (parabolas, hyperbolas, circles, and ellipses), and so on. From there we generalized to get more shapes, which resemble only these old shapes “locally” (we may also think of these new shapes as being “glued” from the old ones). To maintain certain familiar properties expected of shapes, we impose the analogue of the Hausdorff property. We then obtain the concept of a variety.

But we can generalize much, much farther to more than just polynomial rings. We can define “spaces” which come from rings which need not be polynomial rings, such as the ring of ordinary integers $\mathbb{Z}$ (or more generally algebraic integers – we have actually hinted at these applications of algebraic geometry in Divisors and the Picard Group). We can then have a kind of “geometry” of these rings, which gives us methods analogous to the powerful methods of geometry, which can be applied to branches of mathematics we would not usually think of as being “geometric”, such as number theory, as we have mentioned above. We end this post with quotes from two of the pioneers of modern mathematics (these quotes are also found in the book Algebra by Michael Artin):

“To me algebraic geometry is algebra with a kick.”

-Solomon Lefschetz

“In helping geometry, modern algebra is helping itself above all.”

-Oscar Zariski

References:

Algebraic Variety on Wikipedia

Scheme on Wikipedia

Ringed Space on Wikipedia

Abstract Varieties on Rigorous Trivialities

Schemes on Rigorous Trivialities

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

Algebra by Michael Artin